Definition

Definition

A symplectic Lie groupoid is a Lie groupoidX•\mathbf{X}_\bullet whose manifold of morphismsX1\mathbf{X}_1 is equipped with a symplectic structure whose symplectic form ω∈Ωclosed2(X1)\omega \in \Omega^2_{closed}(\mathbf{X}_1) is multiplicative in that the alternating sum of its canonical pullbacks to the space X2\mathbf{X}_2 of composable morphisms vanishes:

Definition

A pre-quasi symplectic groupoid is a Lie groupoidX\mathbf{X} equipped with a differential 2-formω2∈Ω2(X1)\omega_2 \in \Omega^2(\mathbf{X}_1) and a differential 3-form ω3∈Ω3(X0)\omega_3 \in \Omega^3(\mathbf{X}_0) such that

Remark

Since therefore a (pre-)symplectic groupoid is really a Lie groupoid equipped with a cocycle in degree-3 de Rham cohomology (instead of degree 2 as for a symplectic manifold), it is really rather an object in 2-plectic geometry.